The Jones Matrix surface is a simple way to define polarizing components. This article provides some examples of its use.
Authored By Mark Nicholson & Mumlesh Sawasiya
Introduction
Polarization analysis is an extension to conventional ray tracing which considers the effects that optical coatings and reflection and absorption losses have on the propagation of light through a system.
Zemax has detailed analysis capabilities for almost any coating or birefringent medium. However, it is sometimes required to use simpler models, due to a lack of the real prescription data. For example, Zemax supports IDEAL and TABLE coatings for use when real coating data is not available. In a similar manner, the Jones matrix can be used to describe polarization components without doing detailed physical modeling. The Jones matrix is a useful 'black box' approach to modeling some polarization effects.
The Jones Matrix
The amplitude and polarization state of the electric field is described by a vector E which has components {Ex, Ey, Ez} which are all complex-valued. The ray propagation vector k has components {l, m, n} where l, m, and n are the direction cosines of the ray in the x, y and z directions. The electric field vector E must be orthogonal to the propagation vector k so that
and therefore
Any boundary between two media can polarize a beam, and Zemax models this in great detail. However, Zemax also supports an idealized model for a general polarizing device. The model is implemented as a special "Jones Matrix" surface type for Sequential ray tracing, and a "Jones Matrix" object type for Non-Sequential ray tracing. The Jones matrix modifies a Jones vector (which describes the electric field) according to
where A, B, C and D are all complex numbers. In the lens data and in the Non-Sequential components editor, Zemax provides cells for defining A real, A image, etc.
It is important to note that the Jones matrix does not define what happens to the Ez component. This assumes therefore that rays land at normal incidence, i.e. that the idealized polarizer is being placed in a collimated beam. This is a reasonable assumption: most polarizers and waveplates are indeed used in collimated beams or in beams with only small divergence angles.
If the beam is collimated and normal to the Jones matrix, then because k • E = 0 and the vector k has components {0, 0, 1} then Ez must be zero and we can specify the polarization purely in terms of Ex and Ey. If rays land with some arbitrary {l, m, n} then Zemax will adjust either Ez or {Ex, Ey} such that k • E = 0 and the magnitude of E does not increase. The adjustment may however require a reduction in the magnitude of E, and thus an associated loss of transmitted energy.
Here are some typical settings of the Jones matrix coefficients, taken from the Zemax Help System
Off-Axis Incidence to the Jones Matrix
In classical Jones Matrix theory, there is no explicit or universally accepted formulation for handling off-axis incidence. The theory is fundamentally derived for plane waves incident normal to an optical surface, where the polarization basis is well defined. As a result, additional assumptions or mathematical treatments are required when extending the Jones formalism to non-normal (off axis) incidence conditions.
Starting with Zemax version 25R2, off-axis incidence is handled using Rodrigues’ rotation formula, enabling accurate polarization calculations for non-normal incidence. This approach enables a systematic rotation of the electric field vector such that it is effectively transformed into a coordinate system where the field is normally incident on the Jones Matrix. By doing so, the polarization state is correctly aligned with the local reference frame of the optical surface before applying the Jones Matrix operation.
This method ensures a consistent and physically meaningful treatment of polarization effects under off-axis conditions, while maintaining compatibility with the standard Jones formalism. In the following sections, detailed calculation examples are provided to demonstrate the implementation of this approach for both Sequential (SC) and Non-Sequential (NSC) ray-tracing modes.
Rodrigues’ rotation formula describes how to rotate a vector in 3D space about an axis by a given angle.
Rodrigues Rotation Formula (Vector Form)
For a 3D vector v rotated by an angle θ around a unit axis k
Equivalent in Jones Matrix Formalism
For polarization, a rotation of the coordinate system (or optical axis) by an angle θ in the transverse plane (x–y plane) corresponds to a 2×2 rotation matrix acting on the Jones vector:
Applying Rotation to a Jones Matrix
If J is the Jones matrix of an optical element defined in its own coordinate system, and the element is rotated by an angle θ, the rotated Jones matrix J' is given by:
Jones matrix Example in Sequential Mode
Here is an example of a Jones Matrix surface being used as a quarter wave plate. The sample file is included as an attachment to this article.
Note that the Jones Matrix surface does not use the radius of curvature column: it is always a plane. This is consistent with its common use being in collimated light at normal incidence. The matrix elements are entered as parameter data in the Lens Data Editor. In this case, the Jones matrix is configured to act like a quarter-wave-plate in the x-direction:
The easiest way to see the effect of the Jones Matrix surface is with the Polarization Pupil Map, which is located under Analyze...Polarization...Polarization Pupil Map:
It can be seen that the input circular polarization has been altered to a linear polarization, with 100% efficiency.
For Off -Axis Incidence:
When the Jones matrix is rotated by 30 degrees along the X-axis, the output remains unchanged. This shows the Paraxial behavior of the Jones matrix
When Jones Matrix is used as an HWP
If we change the Jones matrix elements to represent a half-wave plate in x (Areal = -1, Dreal = +1, all others zero), we get an output circular polarization with the opposite handedness. Note the direction arrow drawn on the polarization ellipses:
If the Polarizer is rotated by an angle of 30 degree in X-direction, still are getting same result in the Output:
Note: all the analysis features under Analyze...Polarization has settings dialogs that allow the user to enter the input polarization directly. If you use other Analysis features, like say the Huygens PSF, that have a checkbox to 'Use Polarization' but do not explicitly allow you to define the polarization state of the light, the polarization is controlled via a global setting under System Explorer...Polarization.
Jones matrix Example in Non-sequential Mode
Here is an example of a Jones Matrix surface being used as a quarter wave plate. The sample file is included as an attachment to this article.
The handling of Off-axis rays by Jones matrix has been improved. If a ray is not normally incident on the Jones Matrix it is even showing the Paraxial behavior.
In the Non-Sequential file, we have a Source, a Jones Matrix, and six Detectors, all with polarization enabled in the X, Y, and Z directions for both Transmission and Reflection.
In Non-Sequential mode, the Polarization Pupil Map option is not available. Instead, polarization behavior can be analyzed using the Detector Viewer.
After performing ray tracing, we evaluate the total power on the detectors that are flagged for the X, Y, and Z Polarized directions.
The matrix elements are entered as parameter data in the Lens Data Editor. In this case, the Jones matrix is configured to act like a quarter-wave-plate in the x-direction
In Non-Sequential mode, the input polarization needs to be defined by opening the Source Properties, then navigating to Sources → Polarization.
Polarization flag option is used to define the Polarization state of Detector in X, Y and Z direction. If the flag is 1, 2, or 3, the detector will only consider the component of the rays polarized along the local X, Y, and Z directions, respectively, and will consider the phase of the field along those directions as part of the optical path length.
After initial setup, Run the Ray Trace which is located under Analyze...Ray Tracing.
The easiest way to see the effect of Polarization due to Jones Matrix surface is with the Detector Viewer, which is located under Analyze...Detector Viewer
The Six detector has been flagged Polarization along all three axes: X, Y, and Z for Transmitted and Received Intensity.
When linearly polarized light is incident on a quarter-wave plate (QWP), the emerging light becomes circularly polarized. For circular polarization, the electric field components Ex and Ey have equal magnitudes and a phase difference of 90°, while the Ez component is zero. Upon reflection, the circularly polarized light transforms into linearly polarized light as it passes through the quarter-wave plate (QWP). It can be observed that power is detected only in the detector polarized along the Y direction.
The results indicate that the power measured at both the Tx and Ty detectors is equal, which confirms the paraxial characteristics predicted by the Jones matrix. Furthermore, the total detected power of 1 W signifies that the light is completely polarized.
For Off -Axis Incidence to the Jones Matrix
As shown in the image below, the Jones matrix has been rotated by 30° about the X-axis to demonstrate the resulting polarization behavior.
The results indicate that the power measured at both the Tx and Ty detectors is equal, which confirms the Paraxial behavior of the Jones matrix. Furthermore, the total detected power of 1 W signifies that the light is completely polarized
Summary
The Jones Matrix surface in Zemax provides a convenient, idealized model for simulating polarization-dependent optical components when detailed physical or coating data are not available. It applies a 2×2 complex Jones matrix directly to the electric field vector, enabling users to represent elements such as linear polarizers, quarter-wave plates, and half-wave plates in a simplified manner.
While the Jones Matrix surface can be used for both on-axis and off-axis incidence, its behavior remains fundamentally paraxial, meaning that off-axis interactions may not capture all real-world polarization effects. This article analyses how the Jones Matrix surface behaves in both Sequential and Non-Sequential modes for on-axis and off-axis ray incidence. This article describes Zemax features such as the Polarization Pupil Map and Detector Data, which are used to analyze and determine the state of polarization in an optical system